Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $a = \dfrac{7q}{10q^2 + 5q} \div \dfrac{-9}{2q + 1} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{7q}{10q^2 + 5q} \times \dfrac{2q + 1}{-9} $ When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ 7q \times (2q + 1) } { (10q^2 + 5q) \times -9 } $ $ a = \dfrac {7q (2q + 1)} {-9 \times 5q(2q + 1)} $ $ a = \dfrac{7q(2q + 1)}{-45q(2q + 1)} $ We can cancel the $2q + 1$ so long as $2q + 1 \neq 0$ Therefore $q \neq -\dfrac{1}{2}$ $a = \dfrac{7q \cancel{(2q + 1})}{-45q \cancel{(2q + 1)}} = -\dfrac{7q}{45q} = -\dfrac{7}{45} $